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Theorem sbalv 2160
Description: Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
Hypothesis
Ref Expression
sbalv.1 ([𝑦 / 𝑥]𝜑𝜓)
Assertion
Ref Expression
sbalv ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem sbalv
StepHypRef Expression
1 sbal 2159 . 2 ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧[𝑦 / 𝑥]𝜑)
2 sbalv.1 . . 3 ([𝑦 / 𝑥]𝜑𝜓)
32albii 1822 . 2 (∀𝑧[𝑦 / 𝑥]𝜑 ↔ ∀𝑧𝜓)
41, 3bitri 274 1 ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1537  [wsb 2067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-11 2154
This theorem depends on definitions:  df-bi 206  df-ex 1783  df-sb 2068
This theorem is referenced by:  sbex  2278  sbmo  2616  sbabel  2941  sbabelOLD  2942  mo5f  30823
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